Theory of a Decay

It has been known for some time that half-life for a decay, t1/2, can be written in terms of the square root of the a-particle decay energy, Qa, as follows  

This relationship is useful for predicting the expected a-decay half-lives for unknown nuclei. 

The theoretical description of a emission relies on calculating the rate in terms of two factors. The overall rate of emission consists of the product of the rate at which an a particle appears at the inside wall of the nucleus times the (independent) probability that the a particle tunnels through the barrier. Thus, the rate of emission, or the partial decay constant ka, is written as the product of a frequency factor,/, and a transmission coefficient, T, through the barrier  

Some investigators have suggested that this expression should be multiplied by an additional factor to describe the probability of preformation of an a particle inside the parent nucleus. Unfortunately, there is no clear way to calculate such a factor, but empirical estimates have been made. As we will see below, the theoretical estimates of the emission rates are higher than the observed rates, and the preformation factor can be estimated for each measured case. However, there are other uncertainties in the theoretical estimates that contribute to the differences. 

The frequency with which an a particle reaches the edge of a nucleus can be estimated as the velocity divided by the distance across the nucleus. We can take the distance to be twice the radius (something of a maximum value), but the velocity is a little more subtle to estimate. A lower limit for the velocity could be obtained from the kinetic energy of emitted a particle, but the particle is moving inside a potential energy well, and its velocity should be larger and correspond to the well depth plus the external energy. Therefore, the frequency can be written as  

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The matrix elements in angle brackets contain nuclear factors and (in the case of charged particles) the Coulomb barrier penetration probabilities or Gamow factors, originally calculated in the theory of a-decay, which can be roughly estimated as follows (Fig. 2.7). 

Figure 7.7 Plot of the ratio of the calculated partial a-decay half-life for ground-state l = 0 transitions of even-even nuclei to the measured half-lives. The calculations were made using the simple theory of a decay.

Example Problem Calculate the emission rate and half-life for 238U decay from the simple theory of a decay. Compare this to the observed half-life. 

The one-body theory of a decay applies strictiy to even-even a emitters only. The odd-nucleon a emitters, especially in ground-state transitions, decay at a slower rate than that suggested by the simple one-body formulation as applied to even-even nuclei. Consider the data in Figure 7.9 that shows the a-decay half-lives of the even-even and odd A uranium isotopes. The odd A nuclei have substantially longer half-lives than their even-even neighbors do. 

Use the one-body theory of a decay to estimate the half-life of 224Ra for decay by emission of a 14C ion or a 4He ion. The measured half-life for the 14C decay mode is 10-9 relative to the 4He decay mode. Estimate the relative preformation factors for the a particle and 14C nucleus in the parent nuclide. 

Pom and 269110 both decay by the emission of high-energy a particles (Ea =11.6 and 11.1 MeV, respectively). Calculate the expected lifetime of these nuclei using the one-body theory of a decay. The observed half-lives are 45.1 s and 170 jjls, respectively. Comment on any difference between the observed and calculated half-lives. 

Theory of a. decay phenomenon, G. Gamov, R. W. Goumey, E.U. Condon,... 

The microscopic theories of a decay can be divided into several groups. The statistical models were developed in analogy with the statistical theories of nuclear reactions. The a decay is a surface phenomenon, and reliable information on the nuclear surface is available mostly from... 

Several authors(22 30) have contributed to developing the formalism with which the effects of an interface on a dipole inside or near a particle can be treated. In the Rayleigh regime (/ > a), Gersten and Nitzan have made several contributions to the theory of molecular decay rates and energy transfer/22 24) Kerker et alP solved the boundary value problem for a dipole and a spherical particle of arbitrary size, and NcNulty et al.,(26) Ruppin,(27) Chew,(28) and Druger and co-workers(29,30) have used the solution to solve some of the problems of interest. 

Millikan s experiment did not prove, of course, that (he charge on the cathode ray. beta ray, photoelectric, or Zeeman particle was e. But if we call all such particles electrons, and assume that they have e/m = 1.76 x Hi" coulombs/kg. and e = 1.60 x 10" coulomb (and hence m =9.1 x 10 " kg), we find that they fit very well into Bohr s theory of the hydrogen atom and successive, more comprehensive atomic theories, into Richardson s equations for thermionic emission, into Fermi s theory of beta decay, and so on. In other words, a whole web of modem theory and experiment defines the electron. The best current value of e = (1.60206 0.00003) x 10 g coulomb. 

Fermi developed a quantum mechanical theory of (3 decay building on the foundation of the theory for the spontaneous emission of photons by systems in excited states. At first blush these may seem unrelated, but in both cases a system in a very well-defined single state that has excess energy releases the energy spontaneously by the creation of a particle (or particles). The decay constant for the emission of a photon was shown in the appendix E to be given by the general expression ... 

Before the end of 1933 Fermi developed his theory of 0 decay,52 in January 1934 Joliot-Curie discovered the radioactivity induced by a particles,53 and Fermi that induced by neutrons.54 Neutron physics was at the beginning of a new unexpected fast development. 

The principle of small nuclear changes was given a theoretical basis by George Gamow. In 1928 he derived a successful theory of alpha decay, in which the nucleus is quantized and only small particles, such as protons or alpha particles, have a finite probability of tunneling through the nuclear barrier and escaping the nucleus. That... 

As a chair at the University of Rome, Fermi did much of his most important work between 1927 and 1938. Along with the English physicist Paul Dirac but independently, he developed quantum-mechanical statistics that measure particles of half-integer spin (now known as fermions) between 1929 and 1932 he reformulated more simply and elegantly Dirac s then recent work on quantum electronics. In 1933-1934, he published a theory of /3-decay that included what became known as the Fermi interaction, Fermi interactions, and the Fermi coupling constant. Fie also theorized and named the neutrino ( little neutral one ), originally hypothesized by Wolfgang Pauli but not detected experimentally until 1956. 

Statistical methods represent a background for, e.g., the theory of the activated complex (239), the RRKM theory of unimolecular decay (240), the quasi-equilibrium theory of mass spectra (241), and the phase space theory of reaction kinetics (242). These theories yield results in terms of the total reaction cross-sections or detailed macroscopic rate constants. The RRKM and the phase space theory can be obtained as special cases of the single adiabatic channel model (SACM) developed by Quack and Troe (243). The SACM of unimolecular decay provides information on the distribution of the relative kinetic energy of the products released as well as on their angular distributions. 

Grice has derived a microcanonical theory of reactions decaying from a persistent complex [65]. 

There are two common assumptions associated with statistical theories of overall decay, specifically, that the decay is exponential and that, at energy E, it occurs with a statistical lifetime xS(E) = That is, the probability... 

The distribution of a decaying scalar field advected by a turbulent flow was studied by Corrsin (1961) who generalized the Obukhov-Corrsin theory of passive scalar turbulence for the linear decay problem F(C) = S(x) — bC. As in the case of the passive non-decaying scalar field, depending on the length scales considered, one can identify inertial-convective and viscous-convective regimes with qualitatively different characteristics. 

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